3.1 Signed numbers

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of |x|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples.Objectives of this module: be familiar with positive and negative numbers and with the concept of opposites.

Overview

• Positive and Negative Numbers
• Opposites

Positive and negative numbers

When we studied the number line in Section [link] we noted that

Each point on the number line corresponds to a real number, and each real number is located at a unique point on the number line.

Positive and negative numbers

Each real number has a sign inherently associated with it. A real number is said to be a positive number if it is located to the right of 0 on the number line. It is a negative number if it is located to the left of 0 on the number line.

The notation of signed numbers

A number is denoted as positive if it is directly preceded by a $"+"$ sign or no sign at all.
A number is denoted as negative if it is directly preceded by a $"-"$ sign.

The $"+"$ and $"-"$ signs now have two meanings:

$+$ can denote the operation of addition or a positive number.
$-$ can denote the operation of subtraction or a negative number.

Read the $"-"$ Sign as "negative"

To avoid any confusion between "sign" and "operation," it is preferable to read the sign of a number as "positive" or "negative."

Sample set a

$-8$ should be read as "negative eight" rather than "minus eight."

$4+\left(-2\right)$ should be read as "four plus negative two" rather than "four plus minus two."

$-6+\left(-3\right)$ should be read as "negative six plus negative three" rather than "minus six plusminus three."

$-15-\left(-6\right)$ should be read as "negative fifteen minus negative six" rather than "minus fifteenminus minus six."

$-5+7$ should be read as "negative five plus seven" rather than "minus five plus seven."

$0-2$ should be read as "zero minus two."

Practice set a

Write each expression in words.

$4+10$

four plus ten

$7+\left(-4\right)$

seven plus negative four

$-9+2$

negative nine plus two

$-16-\left(+8\right)$

negative sixteen minus positive eight

$-1-\left(-9\right)$

negative one minus negative nine

$0+\left(-7\right)$

zero plus negative seven

Opposites

On the number line, each real number has an image on the opposite side of 0. For this reason we say that each real number has an opposite. Opposites are the same distance from zero but have opposite signs.

The opposite of a real number is denoted by placing a negative sign directly in front of the number. Thus, if $a$ is any real number, then $-a$ is its opposite. Notice that the letter $a$ is a variable. Thus, $"a"$ need not be positive, and $"-a"$ need not be negative.

If $a$ is a real number, $-a$ is opposite $a$ on the number line and $a$ is opposite $-a$ on the number line.

$-\left(-a\right)$ is opposite $-a$ on the number line. This implies that $-\left(-a\right)=a$ .

This property of opposites suggests the double-negative property for real numbers.

The double-negative property

If $a$ is a real number, then
$-\left(-a\right)=a$

Sample set b

If $a=3$ , then $-a=-3$ and $-\left(-a\right)=-\left(-3\right)=3$ .

If $a=-4$ , then $-a=-\left(-4\right)=4$ and $-\left(-a\right)=a=-4$ .

Practice set b

Find the opposite of each real number.

8

$-8$

17

$-17$

$-6$

6

$-15$

15

$-\left(-1\right)$

$-1$ , since $-\left(-1\right)=1$

$-\left[-\left(-7\right)\right]$

7

Suppose that $a$ is a positive number. What type of number is $-a$ ?

If $a$ is positive, $-a$ is negative.

Suppose that $a$ is a negative number. What type of number is $-a$ ?

If $a$ is negative, $-a$ is positive.

Suppose we do not know the sign of the number $m$ . Can we say that $-m$ is positive, negative, or that we do notknow ?

We must say that we do not know.

Exercises

A number is denoted as positive if it is directly preceded by ____________________ .

a plus sign or no sign at all

A number is denoted as negative if it is directly preceded by ____________________ .

For the following problems, how should the real numbers be read ? (Write in words.)

$-5$

a negative five

$-3$

12

twelve

10

$-\left(-4\right)$

negative negative four

$-\left(-1\right)$

For the following problems, write the expressions in words.

$5+7$

five plus seven

$2+6$

$11+\left(-2\right)$

eleven plus negative two

$1+\left(-5\right)$

$6-\left(-8\right)$

six minus negative eight

$0-\left(-15\right)$

Rewrite the following problems in a simpler form.

$-\left(-8\right)$

$-\left(-8\right)=8$

$-\left(-5\right)$

$-\left(-2\right)$

2

$-\left(-9\right)$

$-\left(-1\right)$

1

$-\left(-4\right)$

$-\left[-\left(-3\right)\right]$

$-3$

$-\left[-\left(-10\right)\right]$

$-\left[-\left(-6\right)\right]$

$-6$

$-\left[-\left(-15\right)\right]$

$-\left\{-\left[-\left(-26\right)\right]\right\}$

26

$-\left\{-\left[-\left(-11\right)\right]\right\}$

$-\left\{-\left[-\left(-31\right)\right]\right\}$

31

$-\left\{-\left[-\left(-14\right)\right]\right\}$

$-\left[-\left(12\right)\right]$

12

$-\left[-\left(2\right)\right]$

$-\left[-\left(17\right)\right]$

17

$-\left[-\left(42\right)\right]$

$5-\left(-2\right)$

$5-\left(-2\right)=5+2=7$

$6-\left(-14\right)$

$10-\left(-6\right)$

16

$18-\left(-12\right)$

$31-\left(-1\right)$

32

$54-\left(-18\right)$

$6-\left(-3\right)-\left(-4\right)$

13

$2-\left(-1\right)-\left(-8\right)$

$15-\left(-6\right)-\left(-5\right)$

26

$24-\left(-8\right)-\left(-13\right)$

Exercises for review

( [link] ) There is only one real number for which ${\left(5a\right)}^{2}=5{a}^{2}$ . What is the number?

0

( [link] ) Simplify $\left(3xy\right)\left(2{x}^{2}{y}^{3}\right)\left(4{x}^{2}{y}^{4}\right)$ .

( [link] ) Simplify ${x}^{n+3}\cdot {x}^{5}$ .

${x}^{n+8}$

( [link] ) Simplify ${\left({a}^{3}{b}^{2}{c}^{4}\right)}^{4}$ .

( [link] ) Simplify ${\left(\frac{4{a}^{2}b}{3x{y}^{3}}\right)}^{2}$ .

$\frac{16{a}^{4}{b}^{2}}{9{x}^{2}{y}^{6}}$

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